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ai-content-maker/.venv/Lib/site-packages/sympy/core/evalf.py

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2024-05-03 04:18:51 +03:00
"""
Adaptive numerical evaluation of SymPy expressions, using mpmath
for mathematical functions.
"""
from __future__ import annotations
from typing import Tuple as tTuple, Optional, Union as tUnion, Callable, List, Dict as tDict, Type, TYPE_CHECKING, \
Any, overload
import math
import mpmath.libmp as libmp
from mpmath import (
make_mpc, make_mpf, mp, mpc, mpf, nsum, quadts, quadosc, workprec)
from mpmath import inf as mpmath_inf
from mpmath.libmp import (from_int, from_man_exp, from_rational, fhalf,
fnan, finf, fninf, fnone, fone, fzero, mpf_abs, mpf_add,
mpf_atan, mpf_atan2, mpf_cmp, mpf_cos, mpf_e, mpf_exp, mpf_log, mpf_lt,
mpf_mul, mpf_neg, mpf_pi, mpf_pow, mpf_pow_int, mpf_shift, mpf_sin,
mpf_sqrt, normalize, round_nearest, to_int, to_str)
from mpmath.libmp import bitcount as mpmath_bitcount
from mpmath.libmp.backend import MPZ
from mpmath.libmp.libmpc import _infs_nan
from mpmath.libmp.libmpf import dps_to_prec, prec_to_dps
from .sympify import sympify
from .singleton import S
from sympy.external.gmpy import SYMPY_INTS
from sympy.utilities.iterables import is_sequence
from sympy.utilities.lambdify import lambdify
from sympy.utilities.misc import as_int
if TYPE_CHECKING:
from sympy.core.expr import Expr
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.power import Pow
from sympy.core.symbol import Symbol
from sympy.integrals.integrals import Integral
from sympy.concrete.summations import Sum
from sympy.concrete.products import Product
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.complexes import Abs, re, im
from sympy.functions.elementary.integers import ceiling, floor
from sympy.functions.elementary.trigonometric import atan
from .numbers import Float, Rational, Integer, AlgebraicNumber, Number
LG10 = math.log(10, 2)
rnd = round_nearest
def bitcount(n):
"""Return smallest integer, b, such that |n|/2**b < 1.
"""
return mpmath_bitcount(abs(int(n)))
# Used in a few places as placeholder values to denote exponents and
# precision levels, e.g. of exact numbers. Must be careful to avoid
# passing these to mpmath functions or returning them in final results.
INF = float(mpmath_inf)
MINUS_INF = float(-mpmath_inf)
# ~= 100 digits. Real men set this to INF.
DEFAULT_MAXPREC = 333
class PrecisionExhausted(ArithmeticError):
pass
#----------------------------------------------------------------------------#
# #
# Helper functions for arithmetic and complex parts #
# #
#----------------------------------------------------------------------------#
"""
An mpf value tuple is a tuple of integers (sign, man, exp, bc)
representing a floating-point number: [1, -1][sign]*man*2**exp where
sign is 0 or 1 and bc should correspond to the number of bits used to
represent the mantissa (man) in binary notation, e.g.
"""
MPF_TUP = tTuple[int, int, int, int] # mpf value tuple
"""
Explanation
===========
>>> from sympy.core.evalf import bitcount
>>> sign, man, exp, bc = 0, 5, 1, 3
>>> n = [1, -1][sign]*man*2**exp
>>> n, bitcount(man)
(10, 3)
A temporary result is a tuple (re, im, re_acc, im_acc) where
re and im are nonzero mpf value tuples representing approximate
numbers, or None to denote exact zeros.
re_acc, im_acc are integers denoting log2(e) where e is the estimated
relative accuracy of the respective complex part, but may be anything
if the corresponding complex part is None.
"""
TMP_RES = Any # temporary result, should be some variant of
# tUnion[tTuple[Optional[MPF_TUP], Optional[MPF_TUP],
# Optional[int], Optional[int]],
# 'ComplexInfinity']
# but mypy reports error because it doesn't know as we know
# 1. re and re_acc are either both None or both MPF_TUP
# 2. sometimes the result can't be zoo
# type of the "options" parameter in internal evalf functions
OPT_DICT = tDict[str, Any]
def fastlog(x: Optional[MPF_TUP]) -> tUnion[int, Any]:
"""Fast approximation of log2(x) for an mpf value tuple x.
Explanation
===========
Calculated as exponent + width of mantissa. This is an
approximation for two reasons: 1) it gives the ceil(log2(abs(x)))
value and 2) it is too high by 1 in the case that x is an exact
power of 2. Although this is easy to remedy by testing to see if
the odd mpf mantissa is 1 (indicating that one was dealing with
an exact power of 2) that would decrease the speed and is not
necessary as this is only being used as an approximation for the
number of bits in x. The correct return value could be written as
"x[2] + (x[3] if x[1] != 1 else 0)".
Since mpf tuples always have an odd mantissa, no check is done
to see if the mantissa is a multiple of 2 (in which case the
result would be too large by 1).
Examples
========
>>> from sympy import log
>>> from sympy.core.evalf import fastlog, bitcount
>>> s, m, e = 0, 5, 1
>>> bc = bitcount(m)
>>> n = [1, -1][s]*m*2**e
>>> n, (log(n)/log(2)).evalf(2), fastlog((s, m, e, bc))
(10, 3.3, 4)
"""
if not x or x == fzero:
return MINUS_INF
return x[2] + x[3]
def pure_complex(v: 'Expr', or_real=False) -> tuple['Number', 'Number'] | None:
"""Return a and b if v matches a + I*b where b is not zero and
a and b are Numbers, else None. If `or_real` is True then 0 will
be returned for `b` if `v` is a real number.
Examples
========
>>> from sympy.core.evalf import pure_complex
>>> from sympy import sqrt, I, S
>>> a, b, surd = S(2), S(3), sqrt(2)
>>> pure_complex(a)
>>> pure_complex(a, or_real=True)
(2, 0)
>>> pure_complex(surd)
>>> pure_complex(a + b*I)
(2, 3)
>>> pure_complex(I)
(0, 1)
"""
h, t = v.as_coeff_Add()
if t:
c, i = t.as_coeff_Mul()
if i is S.ImaginaryUnit:
return h, c
elif or_real:
return h, S.Zero
return None
# I don't know what this is, see function scaled_zero below
SCALED_ZERO_TUP = tTuple[List[int], int, int, int]
@overload
def scaled_zero(mag: SCALED_ZERO_TUP, sign=1) -> MPF_TUP:
...
@overload
def scaled_zero(mag: int, sign=1) -> tTuple[SCALED_ZERO_TUP, int]:
...
def scaled_zero(mag: tUnion[SCALED_ZERO_TUP, int], sign=1) -> \
tUnion[MPF_TUP, tTuple[SCALED_ZERO_TUP, int]]:
"""Return an mpf representing a power of two with magnitude ``mag``
and -1 for precision. Or, if ``mag`` is a scaled_zero tuple, then just
remove the sign from within the list that it was initially wrapped
in.
Examples
========
>>> from sympy.core.evalf import scaled_zero
>>> from sympy import Float
>>> z, p = scaled_zero(100)
>>> z, p
(([0], 1, 100, 1), -1)
>>> ok = scaled_zero(z)
>>> ok
(0, 1, 100, 1)
>>> Float(ok)
1.26765060022823e+30
>>> Float(ok, p)
0.e+30
>>> ok, p = scaled_zero(100, -1)
>>> Float(scaled_zero(ok), p)
-0.e+30
"""
if isinstance(mag, tuple) and len(mag) == 4 and iszero(mag, scaled=True):
return (mag[0][0],) + mag[1:]
elif isinstance(mag, SYMPY_INTS):
if sign not in [-1, 1]:
raise ValueError('sign must be +/-1')
rv, p = mpf_shift(fone, mag), -1
s = 0 if sign == 1 else 1
rv = ([s],) + rv[1:]
return rv, p
else:
raise ValueError('scaled zero expects int or scaled_zero tuple.')
def iszero(mpf: tUnion[MPF_TUP, SCALED_ZERO_TUP, None], scaled=False) -> Optional[bool]:
if not scaled:
return not mpf or not mpf[1] and not mpf[-1]
return mpf and isinstance(mpf[0], list) and mpf[1] == mpf[-1] == 1
def complex_accuracy(result: TMP_RES) -> tUnion[int, Any]:
"""
Returns relative accuracy of a complex number with given accuracies
for the real and imaginary parts. The relative accuracy is defined
in the complex norm sense as ||z|+|error|| / |z| where error
is equal to (real absolute error) + (imag absolute error)*i.
The full expression for the (logarithmic) error can be approximated
easily by using the max norm to approximate the complex norm.
In the worst case (re and im equal), this is wrong by a factor
sqrt(2), or by log2(sqrt(2)) = 0.5 bit.
"""
if result is S.ComplexInfinity:
return INF
re, im, re_acc, im_acc = result
if not im:
if not re:
return INF
return re_acc
if not re:
return im_acc
re_size = fastlog(re)
im_size = fastlog(im)
absolute_error = max(re_size - re_acc, im_size - im_acc)
relative_error = absolute_error - max(re_size, im_size)
return -relative_error
def get_abs(expr: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES:
result = evalf(expr, prec + 2, options)
if result is S.ComplexInfinity:
return finf, None, prec, None
re, im, re_acc, im_acc = result
if not re:
re, re_acc, im, im_acc = im, im_acc, re, re_acc
if im:
if expr.is_number:
abs_expr, _, acc, _ = evalf(abs(N(expr, prec + 2)),
prec + 2, options)
return abs_expr, None, acc, None
else:
if 'subs' in options:
return libmp.mpc_abs((re, im), prec), None, re_acc, None
return abs(expr), None, prec, None
elif re:
return mpf_abs(re), None, re_acc, None
else:
return None, None, None, None
def get_complex_part(expr: 'Expr', no: int, prec: int, options: OPT_DICT) -> TMP_RES:
"""no = 0 for real part, no = 1 for imaginary part"""
workprec = prec
i = 0
while 1:
res = evalf(expr, workprec, options)
if res is S.ComplexInfinity:
return fnan, None, prec, None
value, accuracy = res[no::2]
# XXX is the last one correct? Consider re((1+I)**2).n()
if (not value) or accuracy >= prec or -value[2] > prec:
return value, None, accuracy, None
workprec += max(30, 2**i)
i += 1
def evalf_abs(expr: 'Abs', prec: int, options: OPT_DICT) -> TMP_RES:
return get_abs(expr.args[0], prec, options)
def evalf_re(expr: 're', prec: int, options: OPT_DICT) -> TMP_RES:
return get_complex_part(expr.args[0], 0, prec, options)
def evalf_im(expr: 'im', prec: int, options: OPT_DICT) -> TMP_RES:
return get_complex_part(expr.args[0], 1, prec, options)
def finalize_complex(re: MPF_TUP, im: MPF_TUP, prec: int) -> TMP_RES:
if re == fzero and im == fzero:
raise ValueError("got complex zero with unknown accuracy")
elif re == fzero:
return None, im, None, prec
elif im == fzero:
return re, None, prec, None
size_re = fastlog(re)
size_im = fastlog(im)
if size_re > size_im:
re_acc = prec
im_acc = prec + min(-(size_re - size_im), 0)
else:
im_acc = prec
re_acc = prec + min(-(size_im - size_re), 0)
return re, im, re_acc, im_acc
def chop_parts(value: TMP_RES, prec: int) -> TMP_RES:
"""
Chop off tiny real or complex parts.
"""
if value is S.ComplexInfinity:
return value
re, im, re_acc, im_acc = value
# Method 1: chop based on absolute value
if re and re not in _infs_nan and (fastlog(re) < -prec + 4):
re, re_acc = None, None
if im and im not in _infs_nan and (fastlog(im) < -prec + 4):
im, im_acc = None, None
# Method 2: chop if inaccurate and relatively small
if re and im:
delta = fastlog(re) - fastlog(im)
if re_acc < 2 and (delta - re_acc <= -prec + 4):
re, re_acc = None, None
if im_acc < 2 and (delta - im_acc >= prec - 4):
im, im_acc = None, None
return re, im, re_acc, im_acc
def check_target(expr: 'Expr', result: TMP_RES, prec: int):
a = complex_accuracy(result)
if a < prec:
raise PrecisionExhausted("Failed to distinguish the expression: \n\n%s\n\n"
"from zero. Try simplifying the input, using chop=True, or providing "
"a higher maxn for evalf" % (expr))
def get_integer_part(expr: 'Expr', no: int, options: OPT_DICT, return_ints=False) -> \
tUnion[TMP_RES, tTuple[int, int]]:
"""
With no = 1, computes ceiling(expr)
With no = -1, computes floor(expr)
Note: this function either gives the exact result or signals failure.
"""
from sympy.functions.elementary.complexes import re, im
# The expression is likely less than 2^30 or so
assumed_size = 30
result = evalf(expr, assumed_size, options)
if result is S.ComplexInfinity:
raise ValueError("Cannot get integer part of Complex Infinity")
ire, iim, ire_acc, iim_acc = result
# We now know the size, so we can calculate how much extra precision
# (if any) is needed to get within the nearest integer
if ire and iim:
gap = max(fastlog(ire) - ire_acc, fastlog(iim) - iim_acc)
elif ire:
gap = fastlog(ire) - ire_acc
elif iim:
gap = fastlog(iim) - iim_acc
else:
# ... or maybe the expression was exactly zero
if return_ints:
return 0, 0
else:
return None, None, None, None
margin = 10
if gap >= -margin:
prec = margin + assumed_size + gap
ire, iim, ire_acc, iim_acc = evalf(
expr, prec, options)
else:
prec = assumed_size
# We can now easily find the nearest integer, but to find floor/ceil, we
# must also calculate whether the difference to the nearest integer is
# positive or negative (which may fail if very close).
def calc_part(re_im: 'Expr', nexpr: MPF_TUP):
from .add import Add
_, _, exponent, _ = nexpr
is_int = exponent == 0
nint = int(to_int(nexpr, rnd))
if is_int:
# make sure that we had enough precision to distinguish
# between nint and the re or im part (re_im) of expr that
# was passed to calc_part
ire, iim, ire_acc, iim_acc = evalf(
re_im - nint, 10, options) # don't need much precision
assert not iim
size = -fastlog(ire) + 2 # -ve b/c ire is less than 1
if size > prec:
ire, iim, ire_acc, iim_acc = evalf(
re_im, size, options)
assert not iim
nexpr = ire
nint = int(to_int(nexpr, rnd))
_, _, new_exp, _ = ire
is_int = new_exp == 0
if not is_int:
# if there are subs and they all contain integer re/im parts
# then we can (hopefully) safely substitute them into the
# expression
s = options.get('subs', False)
if s:
doit = True
# use strict=False with as_int because we take
# 2.0 == 2
for v in s.values():
try:
as_int(v, strict=False)
except ValueError:
try:
[as_int(i, strict=False) for i in v.as_real_imag()]
continue
except (ValueError, AttributeError):
doit = False
break
if doit:
re_im = re_im.subs(s)
re_im = Add(re_im, -nint, evaluate=False)
x, _, x_acc, _ = evalf(re_im, 10, options)
try:
check_target(re_im, (x, None, x_acc, None), 3)
except PrecisionExhausted:
if not re_im.equals(0):
raise PrecisionExhausted
x = fzero
nint += int(no*(mpf_cmp(x or fzero, fzero) == no))
nint = from_int(nint)
return nint, INF
re_, im_, re_acc, im_acc = None, None, None, None
if ire:
re_, re_acc = calc_part(re(expr, evaluate=False), ire)
if iim:
im_, im_acc = calc_part(im(expr, evaluate=False), iim)
if return_ints:
return int(to_int(re_ or fzero)), int(to_int(im_ or fzero))
return re_, im_, re_acc, im_acc
def evalf_ceiling(expr: 'ceiling', prec: int, options: OPT_DICT) -> TMP_RES:
return get_integer_part(expr.args[0], 1, options)
def evalf_floor(expr: 'floor', prec: int, options: OPT_DICT) -> TMP_RES:
return get_integer_part(expr.args[0], -1, options)
def evalf_float(expr: 'Float', prec: int, options: OPT_DICT) -> TMP_RES:
return expr._mpf_, None, prec, None
def evalf_rational(expr: 'Rational', prec: int, options: OPT_DICT) -> TMP_RES:
return from_rational(expr.p, expr.q, prec), None, prec, None
def evalf_integer(expr: 'Integer', prec: int, options: OPT_DICT) -> TMP_RES:
return from_int(expr.p, prec), None, prec, None
#----------------------------------------------------------------------------#
# #
# Arithmetic operations #
# #
#----------------------------------------------------------------------------#
def add_terms(terms: list, prec: int, target_prec: int) -> \
tTuple[tUnion[MPF_TUP, SCALED_ZERO_TUP, None], Optional[int]]:
"""
Helper for evalf_add. Adds a list of (mpfval, accuracy) terms.
Returns
=======
- None, None if there are no non-zero terms;
- terms[0] if there is only 1 term;
- scaled_zero if the sum of the terms produces a zero by cancellation
e.g. mpfs representing 1 and -1 would produce a scaled zero which need
special handling since they are not actually zero and they are purposely
malformed to ensure that they cannot be used in anything but accuracy
calculations;
- a tuple that is scaled to target_prec that corresponds to the
sum of the terms.
The returned mpf tuple will be normalized to target_prec; the input
prec is used to define the working precision.
XXX explain why this is needed and why one cannot just loop using mpf_add
"""
terms = [t for t in terms if not iszero(t[0])]
if not terms:
return None, None
elif len(terms) == 1:
return terms[0]
# see if any argument is NaN or oo and thus warrants a special return
special = []
from .numbers import Float
for t in terms:
arg = Float._new(t[0], 1)
if arg is S.NaN or arg.is_infinite:
special.append(arg)
if special:
from .add import Add
rv = evalf(Add(*special), prec + 4, {})
return rv[0], rv[2]
working_prec = 2*prec
sum_man, sum_exp = 0, 0
absolute_err: List[int] = []
for x, accuracy in terms:
sign, man, exp, bc = x
if sign:
man = -man
absolute_err.append(bc + exp - accuracy)
delta = exp - sum_exp
if exp >= sum_exp:
# x much larger than existing sum?
# first: quick test
if ((delta > working_prec) and
((not sum_man) or
delta - bitcount(abs(sum_man)) > working_prec)):
sum_man = man
sum_exp = exp
else:
sum_man += (man << delta)
else:
delta = -delta
# x much smaller than existing sum?
if delta - bc > working_prec:
if not sum_man:
sum_man, sum_exp = man, exp
else:
sum_man = (sum_man << delta) + man
sum_exp = exp
absolute_error = max(absolute_err)
if not sum_man:
return scaled_zero(absolute_error)
if sum_man < 0:
sum_sign = 1
sum_man = -sum_man
else:
sum_sign = 0
sum_bc = bitcount(sum_man)
sum_accuracy = sum_exp + sum_bc - absolute_error
r = normalize(sum_sign, sum_man, sum_exp, sum_bc, target_prec,
rnd), sum_accuracy
return r
def evalf_add(v: 'Add', prec: int, options: OPT_DICT) -> TMP_RES:
res = pure_complex(v)
if res:
h, c = res
re, _, re_acc, _ = evalf(h, prec, options)
im, _, im_acc, _ = evalf(c, prec, options)
return re, im, re_acc, im_acc
oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)
i = 0
target_prec = prec
while 1:
options['maxprec'] = min(oldmaxprec, 2*prec)
terms = [evalf(arg, prec + 10, options) for arg in v.args]
n = terms.count(S.ComplexInfinity)
if n >= 2:
return fnan, None, prec, None
re, re_acc = add_terms(
[a[0::2] for a in terms if isinstance(a, tuple) and a[0]], prec, target_prec)
im, im_acc = add_terms(
[a[1::2] for a in terms if isinstance(a, tuple) and a[1]], prec, target_prec)
if n == 1:
if re in (finf, fninf, fnan) or im in (finf, fninf, fnan):
return fnan, None, prec, None
return S.ComplexInfinity
acc = complex_accuracy((re, im, re_acc, im_acc))
if acc >= target_prec:
if options.get('verbose'):
print("ADD: wanted", target_prec, "accurate bits, got", re_acc, im_acc)
break
else:
if (prec - target_prec) > options['maxprec']:
break
prec = prec + max(10 + 2**i, target_prec - acc)
i += 1
if options.get('verbose'):
print("ADD: restarting with prec", prec)
options['maxprec'] = oldmaxprec
if iszero(re, scaled=True):
re = scaled_zero(re)
if iszero(im, scaled=True):
im = scaled_zero(im)
return re, im, re_acc, im_acc
def evalf_mul(v: 'Mul', prec: int, options: OPT_DICT) -> TMP_RES:
res = pure_complex(v)
if res:
# the only pure complex that is a mul is h*I
_, h = res
im, _, im_acc, _ = evalf(h, prec, options)
return None, im, None, im_acc
args = list(v.args)
# see if any argument is NaN or oo and thus warrants a special return
has_zero = False
special = []
from .numbers import Float
for arg in args:
result = evalf(arg, prec, options)
if result is S.ComplexInfinity:
special.append(result)
continue
if result[0] is None:
if result[1] is None:
has_zero = True
continue
num = Float._new(result[0], 1)
if num is S.NaN:
return fnan, None, prec, None
if num.is_infinite:
special.append(num)
if special:
if has_zero:
return fnan, None, prec, None
from .mul import Mul
return evalf(Mul(*special), prec + 4, {})
if has_zero:
return None, None, None, None
# With guard digits, multiplication in the real case does not destroy
# accuracy. This is also true in the complex case when considering the
# total accuracy; however accuracy for the real or imaginary parts
# separately may be lower.
acc = prec
# XXX: big overestimate
working_prec = prec + len(args) + 5
# Empty product is 1
start = man, exp, bc = MPZ(1), 0, 1
# First, we multiply all pure real or pure imaginary numbers.
# direction tells us that the result should be multiplied by
# I**direction; all other numbers get put into complex_factors
# to be multiplied out after the first phase.
last = len(args)
direction = 0
args.append(S.One)
complex_factors = []
for i, arg in enumerate(args):
if i != last and pure_complex(arg):
args[-1] = (args[-1]*arg).expand()
continue
elif i == last and arg is S.One:
continue
re, im, re_acc, im_acc = evalf(arg, working_prec, options)
if re and im:
complex_factors.append((re, im, re_acc, im_acc))
continue
elif re:
(s, m, e, b), w_acc = re, re_acc
elif im:
(s, m, e, b), w_acc = im, im_acc
direction += 1
else:
return None, None, None, None
direction += 2*s
man *= m
exp += e
bc += b
while bc > 3*working_prec:
man >>= working_prec
exp += working_prec
bc -= working_prec
acc = min(acc, w_acc)
sign = (direction & 2) >> 1
if not complex_factors:
v = normalize(sign, man, exp, bitcount(man), prec, rnd)
# multiply by i
if direction & 1:
return None, v, None, acc
else:
return v, None, acc, None
else:
# initialize with the first term
if (man, exp, bc) != start:
# there was a real part; give it an imaginary part
re, im = (sign, man, exp, bitcount(man)), (0, MPZ(0), 0, 0)
i0 = 0
else:
# there is no real part to start (other than the starting 1)
wre, wim, wre_acc, wim_acc = complex_factors[0]
acc = min(acc,
complex_accuracy((wre, wim, wre_acc, wim_acc)))
re = wre
im = wim
i0 = 1
for wre, wim, wre_acc, wim_acc in complex_factors[i0:]:
# acc is the overall accuracy of the product; we aren't
# computing exact accuracies of the product.
acc = min(acc,
complex_accuracy((wre, wim, wre_acc, wim_acc)))
use_prec = working_prec
A = mpf_mul(re, wre, use_prec)
B = mpf_mul(mpf_neg(im), wim, use_prec)
C = mpf_mul(re, wim, use_prec)
D = mpf_mul(im, wre, use_prec)
re = mpf_add(A, B, use_prec)
im = mpf_add(C, D, use_prec)
if options.get('verbose'):
print("MUL: wanted", prec, "accurate bits, got", acc)
# multiply by I
if direction & 1:
re, im = mpf_neg(im), re
return re, im, acc, acc
def evalf_pow(v: 'Pow', prec: int, options) -> TMP_RES:
target_prec = prec
base, exp = v.args
# We handle x**n separately. This has two purposes: 1) it is much
# faster, because we avoid calling evalf on the exponent, and 2) it
# allows better handling of real/imaginary parts that are exactly zero
if exp.is_Integer:
p: int = exp.p # type: ignore
# Exact
if not p:
return fone, None, prec, None
# Exponentiation by p magnifies relative error by |p|, so the
# base must be evaluated with increased precision if p is large
prec += int(math.log(abs(p), 2))
result = evalf(base, prec + 5, options)
if result is S.ComplexInfinity:
if p < 0:
return None, None, None, None
return result
re, im, re_acc, im_acc = result
# Real to integer power
if re and not im:
return mpf_pow_int(re, p, target_prec), None, target_prec, None
# (x*I)**n = I**n * x**n
if im and not re:
z = mpf_pow_int(im, p, target_prec)
case = p % 4
if case == 0:
return z, None, target_prec, None
if case == 1:
return None, z, None, target_prec
if case == 2:
return mpf_neg(z), None, target_prec, None
if case == 3:
return None, mpf_neg(z), None, target_prec
# Zero raised to an integer power
if not re:
if p < 0:
return S.ComplexInfinity
return None, None, None, None
# General complex number to arbitrary integer power
re, im = libmp.mpc_pow_int((re, im), p, prec)
# Assumes full accuracy in input
return finalize_complex(re, im, target_prec)
result = evalf(base, prec + 5, options)
if result is S.ComplexInfinity:
if exp.is_Rational:
if exp < 0:
return None, None, None, None
return result
raise NotImplementedError
# Pure square root
if exp is S.Half:
xre, xim, _, _ = result
# General complex square root
if xim:
re, im = libmp.mpc_sqrt((xre or fzero, xim), prec)
return finalize_complex(re, im, prec)
if not xre:
return None, None, None, None
# Square root of a negative real number
if mpf_lt(xre, fzero):
return None, mpf_sqrt(mpf_neg(xre), prec), None, prec
# Positive square root
return mpf_sqrt(xre, prec), None, prec, None
# We first evaluate the exponent to find its magnitude
# This determines the working precision that must be used
prec += 10
result = evalf(exp, prec, options)
if result is S.ComplexInfinity:
return fnan, None, prec, None
yre, yim, _, _ = result
# Special cases: x**0
if not (yre or yim):
return fone, None, prec, None
ysize = fastlog(yre)
# Restart if too big
# XXX: prec + ysize might exceed maxprec
if ysize > 5:
prec += ysize
yre, yim, _, _ = evalf(exp, prec, options)
# Pure exponential function; no need to evalf the base
if base is S.Exp1:
if yim:
re, im = libmp.mpc_exp((yre or fzero, yim), prec)
return finalize_complex(re, im, target_prec)
return mpf_exp(yre, target_prec), None, target_prec, None
xre, xim, _, _ = evalf(base, prec + 5, options)
# 0**y
if not (xre or xim):
if yim:
return fnan, None, prec, None
if yre[0] == 1: # y < 0
return S.ComplexInfinity
return None, None, None, None
# (real ** complex) or (complex ** complex)
if yim:
re, im = libmp.mpc_pow(
(xre or fzero, xim or fzero), (yre or fzero, yim),
target_prec)
return finalize_complex(re, im, target_prec)
# complex ** real
if xim:
re, im = libmp.mpc_pow_mpf((xre or fzero, xim), yre, target_prec)
return finalize_complex(re, im, target_prec)
# negative ** real
elif mpf_lt(xre, fzero):
re, im = libmp.mpc_pow_mpf((xre, fzero), yre, target_prec)
return finalize_complex(re, im, target_prec)
# positive ** real
else:
return mpf_pow(xre, yre, target_prec), None, target_prec, None
#----------------------------------------------------------------------------#
# #
# Special functions #
# #
#----------------------------------------------------------------------------#
def evalf_exp(expr: 'exp', prec: int, options: OPT_DICT) -> TMP_RES:
from .power import Pow
return evalf_pow(Pow(S.Exp1, expr.exp, evaluate=False), prec, options)
def evalf_trig(v: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES:
"""
This function handles sin and cos of complex arguments.
TODO: should also handle tan of complex arguments.
"""
from sympy.functions.elementary.trigonometric import cos, sin
if isinstance(v, cos):
func = mpf_cos
elif isinstance(v, sin):
func = mpf_sin
else:
raise NotImplementedError
arg = v.args[0]
# 20 extra bits is possibly overkill. It does make the need
# to restart very unlikely
xprec = prec + 20
re, im, re_acc, im_acc = evalf(arg, xprec, options)
if im:
if 'subs' in options:
v = v.subs(options['subs'])
return evalf(v._eval_evalf(prec), prec, options)
if not re:
if isinstance(v, cos):
return fone, None, prec, None
elif isinstance(v, sin):
return None, None, None, None
else:
raise NotImplementedError
# For trigonometric functions, we are interested in the
# fixed-point (absolute) accuracy of the argument.
xsize = fastlog(re)
# Magnitude <= 1.0. OK to compute directly, because there is no
# danger of hitting the first root of cos (with sin, magnitude
# <= 2.0 would actually be ok)
if xsize < 1:
return func(re, prec, rnd), None, prec, None
# Very large
if xsize >= 10:
xprec = prec + xsize
re, im, re_acc, im_acc = evalf(arg, xprec, options)
# Need to repeat in case the argument is very close to a
# multiple of pi (or pi/2), hitting close to a root
while 1:
y = func(re, prec, rnd)
ysize = fastlog(y)
gap = -ysize
accuracy = (xprec - xsize) - gap
if accuracy < prec:
if options.get('verbose'):
print("SIN/COS", accuracy, "wanted", prec, "gap", gap)
print(to_str(y, 10))
if xprec > options.get('maxprec', DEFAULT_MAXPREC):
return y, None, accuracy, None
xprec += gap
re, im, re_acc, im_acc = evalf(arg, xprec, options)
continue
else:
return y, None, prec, None
def evalf_log(expr: 'log', prec: int, options: OPT_DICT) -> TMP_RES:
if len(expr.args)>1:
expr = expr.doit()
return evalf(expr, prec, options)
arg = expr.args[0]
workprec = prec + 10
result = evalf(arg, workprec, options)
if result is S.ComplexInfinity:
return result
xre, xim, xacc, _ = result
# evalf can return NoneTypes if chop=True
# issue 18516, 19623
if xre is xim is None:
# Dear reviewer, I do not know what -inf is;
# it looks to be (1, 0, -789, -3)
# but I'm not sure in general,
# so we just let mpmath figure
# it out by taking log of 0 directly.
# It would be better to return -inf instead.
xre = fzero
if xim:
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import log
# XXX: use get_abs etc instead
re = evalf_log(
log(Abs(arg, evaluate=False), evaluate=False), prec, options)
im = mpf_atan2(xim, xre or fzero, prec)
return re[0], im, re[2], prec
imaginary_term = (mpf_cmp(xre, fzero) < 0)
re = mpf_log(mpf_abs(xre), prec, rnd)
size = fastlog(re)
if prec - size > workprec and re != fzero:
from .add import Add
# We actually need to compute 1+x accurately, not x
add = Add(S.NegativeOne, arg, evaluate=False)
xre, xim, _, _ = evalf_add(add, prec, options)
prec2 = workprec - fastlog(xre)
# xre is now x - 1 so we add 1 back here to calculate x
re = mpf_log(mpf_abs(mpf_add(xre, fone, prec2)), prec, rnd)
re_acc = prec
if imaginary_term:
return re, mpf_pi(prec), re_acc, prec
else:
return re, None, re_acc, None
def evalf_atan(v: 'atan', prec: int, options: OPT_DICT) -> TMP_RES:
arg = v.args[0]
xre, xim, reacc, imacc = evalf(arg, prec + 5, options)
if xre is xim is None:
return (None,)*4
if xim:
raise NotImplementedError
return mpf_atan(xre, prec, rnd), None, prec, None
def evalf_subs(prec: int, subs: dict) -> dict:
""" Change all Float entries in `subs` to have precision prec. """
newsubs = {}
for a, b in subs.items():
b = S(b)
if b.is_Float:
b = b._eval_evalf(prec)
newsubs[a] = b
return newsubs
def evalf_piecewise(expr: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES:
from .numbers import Float, Integer
if 'subs' in options:
expr = expr.subs(evalf_subs(prec, options['subs']))
newopts = options.copy()
del newopts['subs']
if hasattr(expr, 'func'):
return evalf(expr, prec, newopts)
if isinstance(expr, float):
return evalf(Float(expr), prec, newopts)
if isinstance(expr, int):
return evalf(Integer(expr), prec, newopts)
# We still have undefined symbols
raise NotImplementedError
def evalf_alg_num(a: 'AlgebraicNumber', prec: int, options: OPT_DICT) -> TMP_RES:
return evalf(a.to_root(), prec, options)
#----------------------------------------------------------------------------#
# #
# High-level operations #
# #
#----------------------------------------------------------------------------#
def as_mpmath(x: Any, prec: int, options: OPT_DICT) -> tUnion[mpc, mpf]:
from .numbers import Infinity, NegativeInfinity, Zero
x = sympify(x)
if isinstance(x, Zero) or x == 0.0:
return mpf(0)
if isinstance(x, Infinity):
return mpf('inf')
if isinstance(x, NegativeInfinity):
return mpf('-inf')
# XXX
result = evalf(x, prec, options)
return quad_to_mpmath(result)
def do_integral(expr: 'Integral', prec: int, options: OPT_DICT) -> TMP_RES:
func = expr.args[0]
x, xlow, xhigh = expr.args[1]
if xlow == xhigh:
xlow = xhigh = 0
elif x not in func.free_symbols:
# only the difference in limits matters in this case
# so if there is a symbol in common that will cancel
# out when taking the difference, then use that
# difference
if xhigh.free_symbols & xlow.free_symbols:
diff = xhigh - xlow
if diff.is_number:
xlow, xhigh = 0, diff
oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)
options['maxprec'] = min(oldmaxprec, 2*prec)
with workprec(prec + 5):
xlow = as_mpmath(xlow, prec + 15, options)
xhigh = as_mpmath(xhigh, prec + 15, options)
# Integration is like summation, and we can phone home from
# the integrand function to update accuracy summation style
# Note that this accuracy is inaccurate, since it fails
# to account for the variable quadrature weights,
# but it is better than nothing
from sympy.functions.elementary.trigonometric import cos, sin
from .symbol import Wild
have_part = [False, False]
max_real_term: tUnion[float, int] = MINUS_INF
max_imag_term: tUnion[float, int] = MINUS_INF
def f(t: 'Expr') -> tUnion[mpc, mpf]:
nonlocal max_real_term, max_imag_term
re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}})
have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]
max_real_term = max(max_real_term, fastlog(re))
max_imag_term = max(max_imag_term, fastlog(im))
if im:
return mpc(re or fzero, im)
return mpf(re or fzero)
if options.get('quad') == 'osc':
A = Wild('A', exclude=[x])
B = Wild('B', exclude=[x])
D = Wild('D')
m = func.match(cos(A*x + B)*D)
if not m:
m = func.match(sin(A*x + B)*D)
if not m:
raise ValueError("An integrand of the form sin(A*x+B)*f(x) "
"or cos(A*x+B)*f(x) is required for oscillatory quadrature")
period = as_mpmath(2*S.Pi/m[A], prec + 15, options)
result = quadosc(f, [xlow, xhigh], period=period)
# XXX: quadosc does not do error detection yet
quadrature_error = MINUS_INF
else:
result, quadrature_err = quadts(f, [xlow, xhigh], error=1)
quadrature_error = fastlog(quadrature_err._mpf_)
options['maxprec'] = oldmaxprec
if have_part[0]:
re: Optional[MPF_TUP] = result.real._mpf_
re_acc: Optional[int]
if re == fzero:
re_s, re_acc = scaled_zero(int(-max(prec, max_real_term, quadrature_error)))
re = scaled_zero(re_s) # handled ok in evalf_integral
else:
re_acc = int(-max(max_real_term - fastlog(re) - prec, quadrature_error))
else:
re, re_acc = None, None
if have_part[1]:
im: Optional[MPF_TUP] = result.imag._mpf_
im_acc: Optional[int]
if im == fzero:
im_s, im_acc = scaled_zero(int(-max(prec, max_imag_term, quadrature_error)))
im = scaled_zero(im_s) # handled ok in evalf_integral
else:
im_acc = int(-max(max_imag_term - fastlog(im) - prec, quadrature_error))
else:
im, im_acc = None, None
result = re, im, re_acc, im_acc
return result
def evalf_integral(expr: 'Integral', prec: int, options: OPT_DICT) -> TMP_RES:
limits = expr.limits
if len(limits) != 1 or len(limits[0]) != 3:
raise NotImplementedError
workprec = prec
i = 0
maxprec = options.get('maxprec', INF)
while 1:
result = do_integral(expr, workprec, options)
accuracy = complex_accuracy(result)
if accuracy >= prec: # achieved desired precision
break
if workprec >= maxprec: # can't increase accuracy any more
break
if accuracy == -1:
# maybe the answer really is zero and maybe we just haven't increased
# the precision enough. So increase by doubling to not take too long
# to get to maxprec.
workprec *= 2
else:
workprec += max(prec, 2**i)
workprec = min(workprec, maxprec)
i += 1
return result
def check_convergence(numer: 'Expr', denom: 'Expr', n: 'Symbol') -> tTuple[int, Any, Any]:
"""
Returns
=======
(h, g, p) where
-- h is:
> 0 for convergence of rate 1/factorial(n)**h
< 0 for divergence of rate factorial(n)**(-h)
= 0 for geometric or polynomial convergence or divergence
-- abs(g) is:
> 1 for geometric convergence of rate 1/h**n
< 1 for geometric divergence of rate h**n
= 1 for polynomial convergence or divergence
(g < 0 indicates an alternating series)
-- p is:
> 1 for polynomial convergence of rate 1/n**h
<= 1 for polynomial divergence of rate n**(-h)
"""
from sympy.polys.polytools import Poly
npol = Poly(numer, n)
dpol = Poly(denom, n)
p = npol.degree()
q = dpol.degree()
rate = q - p
if rate:
return rate, None, None
constant = dpol.LC() / npol.LC()
from .numbers import equal_valued
if not equal_valued(abs(constant), 1):
return rate, constant, None
if npol.degree() == dpol.degree() == 0:
return rate, constant, 0
pc = npol.all_coeffs()[1]
qc = dpol.all_coeffs()[1]
return rate, constant, (qc - pc)/dpol.LC()
def hypsum(expr: 'Expr', n: 'Symbol', start: int, prec: int) -> mpf:
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from .numbers import Float, equal_valued
from sympy.simplify.simplify import hypersimp
if prec == float('inf'):
raise NotImplementedError('does not support inf prec')
if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()
func1 = lambdify(n, num)
func2 = lambdify(n, den)
h, g, p = check_convergence(num, den, n)
if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))
term = expr.subs(n, 0)
if not term.is_Rational:
raise NotImplementedError("Non rational term functionality is not implemented.")
# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1/g))
if p < 1 or (equal_valued(p, 1) and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
vold = None
ndig = prec_to_dps(prec)
while True:
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process; we check the answer to
# make sure that the desired precision has been reached, too.
prec2 = 4*prec
term0 = (MPZ(term.p) << prec2) // term.q
def summand(k, _term=[term0]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))
with workprec(prec):
v = nsum(summand, [0, mpmath_inf], method='richardson')
vf = Float(v, ndig)
if vold is not None and vold == vf:
break
prec += prec # double precision each time
vold = vf
return v._mpf_
def evalf_prod(expr: 'Product', prec: int, options: OPT_DICT) -> TMP_RES:
if all((l[1] - l[2]).is_Integer for l in expr.limits):
result = evalf(expr.doit(), prec=prec, options=options)
else:
from sympy.concrete.summations import Sum
result = evalf(expr.rewrite(Sum), prec=prec, options=options)
return result
def evalf_sum(expr: 'Sum', prec: int, options: OPT_DICT) -> TMP_RES:
from .numbers import Float
if 'subs' in options:
expr = expr.subs(options['subs'])
func = expr.function
limits = expr.limits
if len(limits) != 1 or len(limits[0]) != 3:
raise NotImplementedError
if func.is_zero:
return None, None, prec, None
prec2 = prec + 10
try:
n, a, b = limits[0]
if b is not S.Infinity or a is S.NegativeInfinity or a != int(a):
raise NotImplementedError
# Use fast hypergeometric summation if possible
v = hypsum(func, n, int(a), prec2)
delta = prec - fastlog(v)
if fastlog(v) < -10:
v = hypsum(func, n, int(a), delta)
return v, None, min(prec, delta), None
except NotImplementedError:
# Euler-Maclaurin summation for general series
eps = Float(2.0)**(-prec)
for i in range(1, 5):
m = n = 2**i * prec
s, err = expr.euler_maclaurin(m=m, n=n, eps=eps,
eval_integral=False)
err = err.evalf()
if err is S.NaN:
raise NotImplementedError
if err <= eps:
break
err = fastlog(evalf(abs(err), 20, options)[0])
re, im, re_acc, im_acc = evalf(s, prec2, options)
if re_acc is None:
re_acc = -err
if im_acc is None:
im_acc = -err
return re, im, re_acc, im_acc
#----------------------------------------------------------------------------#
# #
# Symbolic interface #
# #
#----------------------------------------------------------------------------#
def evalf_symbol(x: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES:
val = options['subs'][x]
if isinstance(val, mpf):
if not val:
return None, None, None, None
return val._mpf_, None, prec, None
else:
if '_cache' not in options:
options['_cache'] = {}
cache = options['_cache']
cached, cached_prec = cache.get(x, (None, MINUS_INF))
if cached_prec >= prec:
return cached
v = evalf(sympify(val), prec, options)
cache[x] = (v, prec)
return v
evalf_table: tDict[Type['Expr'], Callable[['Expr', int, OPT_DICT], TMP_RES]] = {}
def _create_evalf_table():
global evalf_table
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from .add import Add
from .mul import Mul
from .numbers import Exp1, Float, Half, ImaginaryUnit, Integer, NaN, NegativeOne, One, Pi, Rational, \
Zero, ComplexInfinity, AlgebraicNumber
from .power import Pow
from .symbol import Dummy, Symbol
from sympy.functions.elementary.complexes import Abs, im, re
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.integers import ceiling, floor
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import atan, cos, sin
from sympy.integrals.integrals import Integral
evalf_table = {
Symbol: evalf_symbol,
Dummy: evalf_symbol,
Float: evalf_float,
Rational: evalf_rational,
Integer: evalf_integer,
Zero: lambda x, prec, options: (None, None, prec, None),
One: lambda x, prec, options: (fone, None, prec, None),
Half: lambda x, prec, options: (fhalf, None, prec, None),
Pi: lambda x, prec, options: (mpf_pi(prec), None, prec, None),
Exp1: lambda x, prec, options: (mpf_e(prec), None, prec, None),
ImaginaryUnit: lambda x, prec, options: (None, fone, None, prec),
NegativeOne: lambda x, prec, options: (fnone, None, prec, None),
ComplexInfinity: lambda x, prec, options: S.ComplexInfinity,
NaN: lambda x, prec, options: (fnan, None, prec, None),
exp: evalf_exp,
cos: evalf_trig,
sin: evalf_trig,
Add: evalf_add,
Mul: evalf_mul,
Pow: evalf_pow,
log: evalf_log,
atan: evalf_atan,
Abs: evalf_abs,
re: evalf_re,
im: evalf_im,
floor: evalf_floor,
ceiling: evalf_ceiling,
Integral: evalf_integral,
Sum: evalf_sum,
Product: evalf_prod,
Piecewise: evalf_piecewise,
AlgebraicNumber: evalf_alg_num,
}
def evalf(x: 'Expr', prec: int, options: OPT_DICT) -> TMP_RES:
"""
Evaluate the ``Expr`` instance, ``x``
to a binary precision of ``prec``. This
function is supposed to be used internally.
Parameters
==========
x : Expr
The formula to evaluate to a float.
prec : int
The binary precision that the output should have.
options : dict
A dictionary with the same entries as
``EvalfMixin.evalf`` and in addition,
``maxprec`` which is the maximum working precision.
Returns
=======
An optional tuple, ``(re, im, re_acc, im_acc)``
which are the real, imaginary, real accuracy
and imaginary accuracy respectively. ``re`` is
an mpf value tuple and so is ``im``. ``re_acc``
and ``im_acc`` are ints.
NB: all these return values can be ``None``.
If all values are ``None``, then that represents 0.
Note that 0 is also represented as ``fzero = (0, 0, 0, 0)``.
"""
from sympy.functions.elementary.complexes import re as re_, im as im_
try:
rf = evalf_table[type(x)]
r = rf(x, prec, options)
except KeyError:
# Fall back to ordinary evalf if possible
if 'subs' in options:
x = x.subs(evalf_subs(prec, options['subs']))
xe = x._eval_evalf(prec)
if xe is None:
raise NotImplementedError
as_real_imag = getattr(xe, "as_real_imag", None)
if as_real_imag is None:
raise NotImplementedError # e.g. FiniteSet(-1.0, 1.0).evalf()
re, im = as_real_imag()
if re.has(re_) or im.has(im_):
raise NotImplementedError
if re == 0.0:
re = None
reprec = None
elif re.is_number:
re = re._to_mpmath(prec, allow_ints=False)._mpf_
reprec = prec
else:
raise NotImplementedError
if im == 0.0:
im = None
imprec = None
elif im.is_number:
im = im._to_mpmath(prec, allow_ints=False)._mpf_
imprec = prec
else:
raise NotImplementedError
r = re, im, reprec, imprec
if options.get("verbose"):
print("### input", x)
print("### output", to_str(r[0] or fzero, 50) if isinstance(r, tuple) else r)
print("### raw", r) # r[0], r[2]
print()
chop = options.get('chop', False)
if chop:
if chop is True:
chop_prec = prec
else:
# convert (approximately) from given tolerance;
# the formula here will will make 1e-i rounds to 0 for
# i in the range +/-27 while 2e-i will not be chopped
chop_prec = int(round(-3.321*math.log10(chop) + 2.5))
if chop_prec == 3:
chop_prec -= 1
r = chop_parts(r, chop_prec)
if options.get("strict"):
check_target(x, r, prec)
return r
def quad_to_mpmath(q, ctx=None):
"""Turn the quad returned by ``evalf`` into an ``mpf`` or ``mpc``. """
mpc = make_mpc if ctx is None else ctx.make_mpc
mpf = make_mpf if ctx is None else ctx.make_mpf
if q is S.ComplexInfinity:
raise NotImplementedError
re, im, _, _ = q
if im:
if not re:
re = fzero
return mpc((re, im))
elif re:
return mpf(re)
else:
return mpf(fzero)
class EvalfMixin:
"""Mixin class adding evalf capability."""
__slots__ = () # type: tTuple[str, ...]
def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False):
"""
Evaluate the given formula to an accuracy of *n* digits.
Parameters
==========
subs : dict, optional
Substitute numerical values for symbols, e.g.
``subs={x:3, y:1+pi}``. The substitutions must be given as a
dictionary.
maxn : int, optional
Allow a maximum temporary working precision of maxn digits.
chop : bool or number, optional
Specifies how to replace tiny real or imaginary parts in
subresults by exact zeros.
When ``True`` the chop value defaults to standard precision.
Otherwise the chop value is used to determine the
magnitude of "small" for purposes of chopping.
>>> from sympy import N
>>> x = 1e-4
>>> N(x, chop=True)
0.000100000000000000
>>> N(x, chop=1e-5)
0.000100000000000000
>>> N(x, chop=1e-4)
0
strict : bool, optional
Raise ``PrecisionExhausted`` if any subresult fails to
evaluate to full accuracy, given the available maxprec.
quad : str, optional
Choose algorithm for numerical quadrature. By default,
tanh-sinh quadrature is used. For oscillatory
integrals on an infinite interval, try ``quad='osc'``.
verbose : bool, optional
Print debug information.
Notes
=====
When Floats are naively substituted into an expression,
precision errors may adversely affect the result. For example,
adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is
then subtracted, the result will be 0.
That is exactly what happens in the following:
>>> from sympy.abc import x, y, z
>>> values = {x: 1e16, y: 1, z: 1e16}
>>> (x + y - z).subs(values)
0
Using the subs argument for evalf is the accurate way to
evaluate such an expression:
>>> (x + y - z).evalf(subs=values)
1.00000000000000
"""
from .numbers import Float, Number
n = n if n is not None else 15
if subs and is_sequence(subs):
raise TypeError('subs must be given as a dictionary')
# for sake of sage that doesn't like evalf(1)
if n == 1 and isinstance(self, Number):
from .expr import _mag
rv = self.evalf(2, subs, maxn, chop, strict, quad, verbose)
m = _mag(rv)
rv = rv.round(1 - m)
return rv
if not evalf_table:
_create_evalf_table()
prec = dps_to_prec(n)
options = {'maxprec': max(prec, int(maxn*LG10)), 'chop': chop,
'strict': strict, 'verbose': verbose}
if subs is not None:
options['subs'] = subs
if quad is not None:
options['quad'] = quad
try:
result = evalf(self, prec + 4, options)
except NotImplementedError:
# Fall back to the ordinary evalf
if hasattr(self, 'subs') and subs is not None: # issue 20291
v = self.subs(subs)._eval_evalf(prec)
else:
v = self._eval_evalf(prec)
if v is None:
return self
elif not v.is_number:
return v
try:
# If the result is numerical, normalize it
result = evalf(v, prec, options)
except NotImplementedError:
# Probably contains symbols or unknown functions
return v
if result is S.ComplexInfinity:
return result
re, im, re_acc, im_acc = result
if re is S.NaN or im is S.NaN:
return S.NaN
if re:
p = max(min(prec, re_acc), 1)
re = Float._new(re, p)
else:
re = S.Zero
if im:
p = max(min(prec, im_acc), 1)
im = Float._new(im, p)
return re + im*S.ImaginaryUnit
else:
return re
n = evalf
def _evalf(self, prec):
"""Helper for evalf. Does the same thing but takes binary precision"""
r = self._eval_evalf(prec)
if r is None:
r = self
return r
def _eval_evalf(self, prec):
return
def _to_mpmath(self, prec, allow_ints=True):
# mpmath functions accept ints as input
errmsg = "cannot convert to mpmath number"
if allow_ints and self.is_Integer:
return self.p
if hasattr(self, '_as_mpf_val'):
return make_mpf(self._as_mpf_val(prec))
try:
result = evalf(self, prec, {})
return quad_to_mpmath(result)
except NotImplementedError:
v = self._eval_evalf(prec)
if v is None:
raise ValueError(errmsg)
if v.is_Float:
return make_mpf(v._mpf_)
# Number + Number*I is also fine
re, im = v.as_real_imag()
if allow_ints and re.is_Integer:
re = from_int(re.p)
elif re.is_Float:
re = re._mpf_
else:
raise ValueError(errmsg)
if allow_ints and im.is_Integer:
im = from_int(im.p)
elif im.is_Float:
im = im._mpf_
else:
raise ValueError(errmsg)
return make_mpc((re, im))
def N(x, n=15, **options):
r"""
Calls x.evalf(n, \*\*options).
Explanations
============
Both .n() and N() are equivalent to .evalf(); use the one that you like better.
See also the docstring of .evalf() for information on the options.
Examples
========
>>> from sympy import Sum, oo, N
>>> from sympy.abc import k
>>> Sum(1/k**k, (k, 1, oo))
Sum(k**(-k), (k, 1, oo))
>>> N(_, 4)
1.291
"""
# by using rational=True, any evaluation of a string
# will be done using exact values for the Floats
return sympify(x, rational=True).evalf(n, **options)
def _evalf_with_bounded_error(x: 'Expr', eps: 'Optional[Expr]' = None,
m: int = 0,
options: Optional[OPT_DICT] = None) -> TMP_RES:
"""
Evaluate *x* to within a bounded absolute error.
Parameters
==========
x : Expr
The quantity to be evaluated.
eps : Expr, None, optional (default=None)
Positive real upper bound on the acceptable error.
m : int, optional (default=0)
If *eps* is None, then use 2**(-m) as the upper bound on the error.
options: OPT_DICT
As in the ``evalf`` function.
Returns
=======
A tuple ``(re, im, re_acc, im_acc)``, as returned by ``evalf``.
See Also
========
evalf
"""
if eps is not None:
if not (eps.is_Rational or eps.is_Float) or not eps > 0:
raise ValueError("eps must be positive")
r, _, _, _ = evalf(1/eps, 1, {})
m = fastlog(r)
c, d, _, _ = evalf(x, 1, {})
# Note: If x = a + b*I, then |a| <= 2|c| and |b| <= 2|d|, with equality
# only in the zero case.
# If a is non-zero, then |c| = 2**nc for some integer nc, and c has
# bitcount 1. Therefore 2**fastlog(c) = 2**(nc+1) = 2|c| is an upper bound
# on |a|. Likewise for b and d.
nr, ni = fastlog(c), fastlog(d)
n = max(nr, ni) + 1
# If x is 0, then n is MINUS_INF, and p will be 1. Otherwise,
# n - 1 bits get us past the integer parts of a and b, and +1 accounts for
# the factor of <= sqrt(2) that is |x|/max(|a|, |b|).
p = max(1, m + n + 1)
options = options or {}
return evalf(x, p, options)